92 
hence, denoting the absolute normal velocity by N, the 
following equation obtains, viz. — 
N — rucosy - VCOSa 2 
4. The normal velocity then as expressed by equation (2) 
can be made to assume different values by assuming different 
values of the available angles a, v which determine the posi- 
tion of the normal line to the element (a) of the propeller 
blade. 
The normal velocity of the element ( a ) will be zero, when 
the following relation between the angular velocity (u) and 
the velocity of translation (V) obtains, viz. 
rucos.y o 
* Hill •e©ooo*oooooooo£aoo»oo«eo O 
COS. a 
This equation is derived directly from equation 2. 
It appears, therefore, that when the element (a) moves 
in accordance with the relation (3) it produces no resistance 
whatever either to rotation or translation along the axis of 
the propeller. In this case the element (a) moves upon 
a surface which may be not inappropriately denominated 
the surface of vanishing pressure. 
5. With reference to equation (1) Art. I., it will be readily 
seen that the condition a,v being complimentary (that is, 
a + v = ^ will give (3 = zero. Hence, in this case, the normal 
line of the element {a) placed at P is situated in a plane at 
right angles to the radius (r). Further, if each of the two 
angles a,r remains constant for the same value of (r) as well 
as satisfy the above condition, then the curve of the pro- 
peller at (r) distance from the axis will be the common 
helix of the Archimedean screw as used in Her Majesty’s 
Navy. 
6. In the determination of the formula 2 Art. 3 it was 
assumed that the element (a), with a normal velocity N, 
struck the water when perfectly at rest. The position, 
however, of the propeller practically placed at the stern of 
